Method of generating helical gears



May 15, 1928.

1,669,919 N. TRBOJEVICH METHOD OF GENERATING HELICAL GEARS Filed July 21, 1924 4 Sheets-Sheet 1 Snuentor ya/evaa/ r'r anba.

May 15, 1928. 1,669,919

N. TRBOJEVICH METHOD OF GENERATINQ HELICAL GEARS Filed July 21, 1924 4 Sheets-Sheet 2 T 1 )V WWW May 15, 1928. 1,669,919

N. TRBOJEVICH METHOD OF GENERATING HELICAL GEARS Filed July 'I2i, 1924 4 Sheets-Sheet 3 luvs May 15, 1928.

1,669,919 N. TRBOJEVICH METHOD OF GENERATING HELICAL GEARS Filed July 21, 1924 4 Sheets-Sheet 4 rolling the blank Patented May 15,

M UNITED STA.

NIKOLA TRBOJ'EVICH, OF HIGHLAND PARK, MICHIGAN.

The invention relates to a novel method of gear generating 'wh plicable to grinding of helical gears.

process are that it is ex tremely simple and also theoretically accuadvantages of the rate, as true involute now be produced by the combination of a plane disk wheel and a screw motion of the blank. alone. In that unique, as in no other known in the art is it possible to generate involute surfaces with against said wheel.

ing element is wholly cable limit (on accou grinding wheel) bein ally detract from the This does not materi value of the invention,

gears a gre now generally e invention,

more accurately and, straight toothed gears, and itis not unreasonable to expect that the straight teeth in gears will be gradually replaced by helical teeth in the future.

In the drawings diagrams have been in a transverse plane In my method the rollater helical angle than 20 degrees,

second, owing to the peculiar result of tl'liS helical ears mav now be round METHOD OF GENERATING BELIGAL GEARS.

Application filed July 21, 1.924. Serial "No. 727,324.

ich is particularly aphelical surfaces may mathematical and been knownin the respect the method 15 gear grmding process curve, they face ofthe curve,

a plane wheel without absent; however, only particular,

nt of the size of the helical gear tooth.

0 about 20 degrees.

as it will now as first, the helical mployed usually have a, parametric form, and

economically than defined as follows:

are 8 as the parameter. equations 1n w, y, z of the following form:

=f1 Zffz where f,, f and 7, denote three functions Fig. 12 is a diagram showin the method of trimming the plane face of the grinder The principle upon which this method of helical gear grinding is ,based is wholly relates to the theory of developable twisted surfaces. It has long higher mathematics that if a series of tangents are drawn to a twisted form the'so called tangential surwhich surface has the unique property among all other twisted or warped surfaces that it is developable into a plane without tearing or crumpling.

I have found that the tangential surface to a circular helix is "identical with the tooth surface of the well known involute This fact may be mathematically proved be briefly indicated. Let us first write down the equations of a. helix in selecting the length of This leads to three It is now evident that the tangent to the helix, expressed by equations (1) may be 0w. 0 th n denoting the length of the tangent, from P and the prime indexes changes from the quoted work by Schefl'ers. Fig. 6 is a plan view of a common helical (8) (8) gear. 3 3

Fig. 7 shows the plan elevation of the same gear in intersection with the rectifying plane.

Fig. 8 is a view of Fig. 7 taken in the norto P (see Figure 1),

mal plane NN.

new method.

Fig. his a detail showing the plan view of the grinder head.

in the quantities f' entiation of m, y, 2,

Let us now assume an X Y Z coordinate (s) etc. denote a differ-- with respect to s.

that the helix h (Figure 2) iswrapped upon radius a (the base cylinder of thegear); that the helical angle is equal to y and that the cylinder of a system is so selected that the axis Z coincides with the axis of about said cylinder.

, base circle a.

the base cylinder and of the base h, the plane X Y. being perpendicular to said axis. In that case the equations (2) assume the following form: a

s sin s sin X=a cos 1' S111 'y Sm s sin s sin 1' Y=a s1n +r s1n '7 cos- (3) Z=s cos 7 +1- cos (The equations (3), except for the difference in notation, may be found in Geo. Scheffers, Differential Geometry, vol. 1, page 359.) '7 a (The equations (3) furnish a mathematical proof that the surface therein defined is identical with the common involute helical tooth surface). Because, the common helical tooth surface is defined as being always an involute of a circle in any plane section perpendicular to the gear axis, and a circuar helix in any cylindrical section concentrio with said axis. It is readily seen that 0 the surface (3) satisfies both above conditions. First, a plane Z=O is perpendicular to the Z axis. and by substituting that value in the last of the equations (3) we have If we now replace the last found value of 1' in the first two equations, the locus is undoubtedly an involute drawn from the On the other hand, a cylindrical section is obtained of the surface (3) if the value -r=const (5) is substituted. The result is a circular helix, drawn upon a cylinder of a radius Referring now again to Fig. 2, the generation of a helical tooth surface by means of a moving tangent along a helix will be understood. Let the plane B be tangent to the base cylinder of aradius a, which plane we shall term as the base plane of the gear (in mathematics the plane is known as the rectifying plane) A straight line t lying in the plane B is inclined with respect to the axis Z at an angle yo (the base helix angle) and touches the cylinder at the point. If the plane B be now rolled without slipping upon thebase cylinder always maintaining its parallel position with respect to the axis Z, the line 2! will describe a helix [2. upon the base cylinder, and it also will describe a twisted surface composed of two nappes The plane sections of the generated surface (parallel to the plane XY) are all involutes such as (Z described by the point A. That the curve d is an involute is easily seen, because the projection 6 of the tangent 15 rolls without slipping upon, and in the plane of the circle 0 The involute described by the opposite point A is also an involute similar to d except that it is of an opposite hand. This fact, as we shall see, is of practical importance because in a gear tooth, the opposite sides are both involutes, one being wound clockwise, and the other counterclockwise. Thus, by the simple expedient of offsetting the generating element from one side of the base helix to the other (from the point A to A) first one and then the other side of a tooth may be finished.

The tangent t to the involute'd (Fig. 2) is perpendicular to both lines t and t, and is parallel to the line DO=n, the principal normal of the helix it. The plane composed of the three intersecting lines 11, t and t is the so called osculating plane of the base helix k, and is tangent to the generated helical surface along the entire length of the line It. Further, said plane is per endicular to the plane B. This, again is 0 practical importance because in this method of gear grinding the plane of the grinding wheel always lies in the osculating plane T of the base helix and thus envelops or generates the surface by touching it along the generator t.

In Figure 3 the two nappes of the tangential surface are shown in perspective. The plane (n is tangent to both nappes along the line AA= Said plane crosses or intersects the surface at the point D, but it does not intersect said surface anywhere else. The normal plane erected at the point D to the base helix is perpendicular to both the osculating and the rectifying planes and intersects the surface in curves e and e which curves although somewhat similar to, are not involutes.

Fig. 4 shows a portion of an involute heliin perspective. That diagram will be easily understood from what already has been disclosed. The osculating plane T is shown in full view in the form of a small parallelogram at the left lower side of the diagram. It is significant that all osculating planes, no matter at which point of a helical tooth surface they are drawn, intersect the axis of the gear at a constant angle y which angle, however, is different from the commonly known and understood pitch helix angle y. This forms an important factor in my discovery, as I am the first to generate helical gears on the osculating plane principle.

The surface shown in Fig. 3 and Fig. 4 is developable into a plane. This fact is well established in mathematics, and is also more or less obvious from Fig. 5. Let the consecutive osculating from each other only distance of angle) be T,

planes (separated by an infinitesimal T T etc. According to the supposition each of those planes contains the corresponding tangent or generator t, t 15,, in their full length. On the other hand, the plane T, intersects the consecut ve plane T in the line t,. Therefore, the lines t, and t lying in the same plane T intersect each other at P and they may be developed into any desired plane without tearing or crumpling ing the generator t, or t,, or both, about the point of intersection P until they coincide with the plane of development.

The prac ical application of the above theoretical considerations will now be shown. In Fig. 6 and Fig. 7 two views fof a common helical gear (right handed) are shown. In the transverse section (Fig. 6) a number of similar equispaced teeth are arranged circumferentially along the pitch circle 0, said teeth being bounded by the left hand involutes (l at one side, and the right hand involutes cl on the other side. All involutes are drawn from the same base circle 0 having a radius a. If we now denote the pitch radius OE =1', and the angle DOE =a, (the transverse pressure angle), we have this fundamental relation:

(1 COS a Let us now intersect the gear G with the rectifying or the base plane B drawn through the point D. This plane intersects the involutes d at right angles at the points E, E etc. and the involutes d on the opposite side of the center line OD, also at right angles, at the points F, F etc. Further, the spacing of the points E, E E E =F, F =F, F,=p is constant and is equal to the base pitch, its exact value bep =p cos a (7 where p is the (measured along the pitch circle 0).

It is now evident that the helical teeth when intersected with the base plane will show a strai ht line of intersection on their sides, lying furthest from the center line OD. Because, as already illustrated in Fig. 2, the base plane B intersects the involutes d at right angles and contains the straight line generator t passing through the point A, in full. Said line-t lies wholly on the tooth surface and is inclined with respect to the axis Z at an angle In Fig. 7 the line t=H H, is shown in its true length andthis length may be determined if the length H, H, (Fig. 6) intercepted in projection by the root circle a, and the top or outside circle 0 of the gear is known, as well as the angle 7 Thus.

The normal base pitch P (the distance the surface by rotatalways contains the line If, and

circular pitch of the gearpitch 25 helix angle, 14

teeth) may also be determined: -I o o cos Y0 the lead of the helix (either of helix) be denoted Let now the pitch helix .or the base with L. Then the pitch helix angle is evidently:

' tan L (10) and thebase helix angle: tan (11) or, from (6) tan =tan 7 cos 0: (12) The above equation (12) is of fundamental importance in this process as the grinding wheel J (Fig. 7) must be inclined to the angle y with respect to the gear axis.

Thus, the new method of gear grinding may now be defined. The line t= 1-I, H Fig. 7 is evidently the trace of the osculating plane in the plane of paper as that plane is perpendicular tothe rectifying plane B (the plane of paper.) 'The line t enters, therefore, the tooth surface at the crosses the pitch line at an acute angle d at the pitch point E and emerges on the top of the tooth at H,. This condition is shown in detail in Fig. 8'which is a view of Fig. 7 taken in the normal plane NN. The object now is to replace the osculating plane T with the grinding wheel J. This may be satisfactorily done providing the grinder J is of a sufliciently large diameter so as to include the whole length of the line H, I-I as its chord, and is placed so with respect to the gear (offset to one side) that its outer circumference will clear the bottoms of the interdental spaces, and its conical rear side (as shown in section at the left of the Fig. 8) will not interfere with the adjacent tooth. Ina practical example, I have ground timing gears used in a well known automobile, 66 teeth, 10 normal diametral pressure angle, 1 face, with a wheel of only 7 diameter. The required diameter of the wheel as is seen from the equation (8) rapidly increases however as the angle y decreases, and for =O (spur gears) t=infinite. For that reason spur gears cannot be ground on this principle.

Suppose now that the grinder J (Figure 8) touchesthe gear tooth along the straight line H, H',. The grinder is rapidly rotated on its axis in a fixed position, and the blank G, Fig. 7, is moved in a helical path,

each revolution about the axis the blank will bottom of tooth at H',,

' upon the tooth surface advance an axial distance L equal to the lead of helix. In this manner all the consecutive generators at lying on the same tooth surface come progressively into contact with the grinder, and as it might be described, develop themselves into the plane represented by the rotary grinder and are thus ground over. This will be better understood from Figure 9 which shows the conditions existing in Figure 8 when the quadrilateral tooth surface bounded by the helixes c, and c and the two opposite involutes (l is developed into .a plane. In that case the helixes c, and 0, become concentric circles, the two involutes (I become the parallel involutes d drawn from the base circle having a radius a equal to the normal radius of curvature of the base helix It, that is:

and the straight line generators 25 all remain straight lines tangent to the new base circle a, and intersecting the pitch helix 0 at an acute angle a lVhen the blank G, Figure 7, is given ascrew motion, in development (F igure 9) it will appear as if the development itself was rotated about the axis Z. The grinding wheel, however, does not participate 1n this rotation, as its line of action H H (Figure 9) remains stationary with respect to said dQVElOPIHQlll) and therefore describes a series of diverging straight lines, all tangent to the developed base circle a,

d 0 d 0 as indicated by shaded area in Figure 9. It is seen that in this manner gears having any desired width of face may be correctly generated.

\Vhen one side of a tooth has thus been generated, the blank is indexed, that is disengaged from its driving arbor and turned through an angle corresponding to the spacing of teeth. and the process is so repeated until all teeth are ground on their one side. To grind the other side either the blank is removed from the arbor and reversed, or the grinder is reversed and moved to the position J Figure 7, or again, two or more grinders may be mounted upon the same arbor, and if they are correctly spaced from each other, they will simultaneously grind the teeth upon their both sides.

A front elevation of a machine which may be used in connection with this method is shown in Figure 10, Figure 11 is a plan view showing the method of,placing the grinder relative to the blank, and Figure 12 shows the method of wheel dressing by means of a trimming diamond. A machine of this kind to be operative must contain the following elements: Means for translating the blank in a helical path, means for indexing, means for angularly adjusting the grinder in a plane parallel to gear axis, means for offsetting the wheel, and means for trimming the wheel to a true plane surface perpendicular to the axis of rotation of said wheel.

Upon the base 21 (Figure 10) a stationary nut 22 is mounted engaging the master screw 23 having the exact lead L and the same hand of helix which it is desired to generate upon blanks 24 and 24. A longitudinally slidable table is mounted in suitable ways upon said base 21 and may be reciprocatcd by turning the hand lever 26 keyed to the pinion 27, said pinion engaging a rack 27' at the bottom of the slide 25. The master screw 23 is rotatably housed in the hear ing 28 integral with the slide 25 and the end play is prevented by means of the thrust collars .29 and 30. At the right hand extremity of the master screw 23 a disk 31 is keyed, carrying the index pin 32. The index plate 33 having a number of equally spaced holes or notches corresponding to the number of teeth which it is desired to generate is rotatably mounted upon the boss integral with the disk 31, and may thus be disen: gaged from said disk by withdrawing the index pin 32. The blanks 24 and 24, are mounted upon the arbor 34 suspended between the centers 35 and 36, the latter beinghoused in the tail stock 37. A dog 38 serves to transmit the rotation from the screw 23 to the blank arbor 34 and. by virtue of'the set screw 39 it also serves for imparting a fine rotative adjustment to the blanks relative to. the wheel.

The grinder 40 (Figure 11) faces the gear blanks with its accurately trimmed plane surface, and is tilted with respect to the axis of the,blank to the base helix angle y while its lowestpoint is at a distance K (in vertical projection) from said gear axis. The distance K may be determined either graphically or by calculation and corresponds to the distance of the point E from the Z axis (see Figure 7). The grinder 40 is mounted upon a suitable spindle which is driven by means of the pulley 41 and is housed in two suitable bearings 42 and 43. The base of the grinding head is rotatable in circular ways, and is adjustable relative to the dial 44 to any exact angle 7 The plane side of the grinder 40 is periodically trimmed to its exact contour by reciprocating the trimmer diamond 45 across the face of wheel while the wheel is rapidly rotating, as will be understood from Figure 12.

lUu

As previously stated. the production of the machine shown in Figure 10 may be increased by mounting two or more grinders upon the grinder spindle, such as the grinders J and J in Figure 7. It is also obvious that instead of grinders,-milling cutters may be employed. It should be noted, however, that if several cutters are simultaneously employed. their outer diameters must be so selected and correlated that they will all cut to the same depth in the gear blank. They also must be correctly spaced, the spacing depending upon the dimension p,,, the number of teeth, helix angle, etc. of the blank, as shown in Figure 7.

What I claim as my invention is 1. The method of grinding helical gears consisting in arranging a plane disk grinder to contact with the gear tooth so that the straight line generator of the twisted helical tooth surface will appear as a chord fully lying in said grinder and lying in a plane parallel to the axes of both the cutter and the grinder and in imparting to the blank a relative movement such as to develop the successive generators progressively into the cutting plane of said grinder.

2. A method of generating involute helical gear teeth consisting in mounting a gear blank upon a rotary arbor, in selecting a rotary plane disk cutter whose cutting plane is perpendicular to its axis of rotation, in placing said cutter in a position coincidingwith the osculating plane of the imaginary base helix of which the tooth surface is the tangential developable, in outwardly offsetting the cutter in said osculating plane until it contacts with the side of the helical tooth lying farthest from the center line of gear and until it includes as its chord the portion of the generating straight line extending obliquely across and from top to the bottom of said surface, in imparting to the blank a relative helical movement concentric with its axis and in continuing with said helical movement until the tooth is finished in the entire length of its face and at all its bearing points.

3. Amethod of generating helical teeth in a gear blank by means of a plane rotary disk cutter in which the position of the cutter relative to the blank is determined by first drawing a rectifying plane tangent-to the imaginary base cylinder of the gear, in finding the intersection of said rectifying plane with the gear tooth which is a straight line, in placing the cutting plane of the cutter in a. perpendicular position relative to the rectifying plane so that it will include the portion of the generating straight line lying on the gear tooth, as its chord, in rotating and translating the blank along its axis in a helical path until the successive straight line generators lying on the same tooth surface are all finished, and in finishing the rest of the gear teeth by indexing and repeating the helical movement.

4. A method of generating twisted surfaces which are tangential developables of a base curve consisting in selecting a plane rotary disk cutter, in aligning its cutting plane with the momentaryosculating plane to the curve and offsetting said cutting plane in the osculating plane until the line to be generated appears as a chord drawn in the cutting circle, in imparting to the surface a relative twist so proportioned that the cutting plane always osculates the base curve, and in continuing said twisting movement until all straight lines lying appearing upon the surface are fully generated.

5. A method of generating helical gears consisting in mounting the gear blank upon the work arbor and two plane disk cutters upon the cutter arbor, the arran ement being such that the cutter arbor ies in the normal plane drawn to the base helix, and the plane side of each cutter faces the other cutter and the center line of the gear and coincides with the osculating planes drawn to the corresponding tooth surfaces acted .upon, the line of contact in each case appearing as a chord drawn in the cutting circle of the corresponding cutter, in imparting to the blank a relative movement of translation along its axis and in a helical path, and in indexing the blank to repeat the cut at another portion of the circumference of the blank.

6. A method of generating twisted surfaces which are tangential developables of a base curve consisting in mounting the gear blank upon the work arbor and two plane disk cutters upon the cutter arbor, the arrangement being such that the cutter arbor lies in the normal plane drawn to the base helix, and the plane side of each cutter faces the other cutter and the center line of the gear and coincides with the osculating planes drawn to the corresponding tooth surfaces acted upon, the-line of contact in each case appearing as a chord drawn in the cutting circle of the corresponding cutter, in imparting. to the blank -a relative twist so proportioned that the cutting planes always osculate the base curves and in indexing the blank to repeat the cut at another position of the circumference of the blank.

In testimony whereof I aflix my signature.

NIKGLA TRBOJEVICH. 

